Asymptotic optimality
of estimating functions
Efficient estimation (cf. Efficient estimator) of parameters in stochastic models is most conveniently approached via properties of estimating functions, namely functions of the data and the parameter of interest, rather than estimators derived therefrom. For a detailed explanation see [a1], Chapt. 1.
Let be a sample in discrete or continuous time from a stochastic system taking values in an
-dimensional Euclidean space. The distribution of
depends on a parameter of interest
taking values in an open subset of a
-dimensional Euclidean space. The possible probability measures (cf. Probability measure) for
are
, a union of families of models.
Consider the class of zero-mean square-integrable estimating functions
, which are vectors of dimension
and for which the matrices used below are non-singular.
Optimality in both the fixed sample and the asymptotic sense is considered. The former involves choice of an estimating function to maximize, in the partial order of non-negative definite matrices, the information criterion
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which is a natural generalization of the Fisher amount of information. Here is the
-matrix of derivatives of the elements of
with respect to those of
and prime denotes transposition. If
is a prespecified family of estimating functions, it is said that
is fixed sample optimal in
if
is non-negative definite for all
,
and
. Then,
is the element of
whose dispersion distance from the maximum information estimating function in
(often the likelihood score) is least.
A focus on asymptotic properties can be made by confining attention to the subset of estimating functions which are martingales (cf. Martingale). Here one considers
ranging over the positive real numbers and for
one writes
for the quadratic characteristic, the predictable increasing process for which
is a martingale. Also, write
for the predictable process for which
is a martingale. Then,
is asymptotically optimal in
if
is almost surely non-negative definite for all
,
,
, and
, where
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Under suitable regularity conditions, asymptotically optimal estimating functions produce estimators for which are consistent (cf. Consistent estimator), asymptotically unbiased (cf. Unbiased estimator) and asymptotically normally distributed (cf. Normal distribution) with minimum size asymptotic confidence zones (cf. Confidence estimation). For further details see [a2], [a3].
References
[a1] | D.L. McLeish, C.G. Small, "The theory and applications of statistical inference functions" , Lecture Notes in Statistics , Springer (1988) |
[a2] | V.P. Godambe, C.C. Heyde, "Quasi-likelihood and optimal estimation" Internat. Statist. Rev. , 55 (1987) pp. 231–244. |
[a3] | C.C. Heyde, "Quasi-likelihood and its application. A general approach to optimal parameter estimation" , Springer (1997) |
Asymptotic optimality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_optimality&oldid=45243